Motion in Brane World Models: The Bazanski Approach

Abstract

Recently, path equations have been obtained for charged, spinning objects in brane world models, using a modified Bazanski Lagrangian. In this study, path deviation equations of extended objects are derived. The significance of moving extended objects in brane world models is examined. Motion in non- symmetric brane world models is also considered.

Full text

arXiv:0708.2015v2 [gr-qc] 20 Aug 2007
Motion in Brane World Models: The Bazanski Approach
M.E.Kahil
1
Abstract
Recently, path equations have been obtained for charged, spinning objects in
brane world models, using a modified Bazanski Lagrangian. In this study, path
deviation equations of extended objects are derived. The significance of movin g
extended objects in brane world models is examined. Motion in non- symmetric
brane world mo dels is also considered.
The Bazanski Approach in Riemannian Geometry
Geodesic and geodesic deviation equations can be obtained simultaneously by ap-
plying the action principle on the Bazanski Lagrangian [1]:
L = g
αβ
U
α
DΨ
β
Ds
, (1)
where
D
D s
is the covariant derivative. By taking the variation with respect to the
deviation vector Ψ
ρ
one obtains the geodesic equation. Taking the variation with
respect to the unit tangent vector U
ρ
, one obtains its geodesic deviation equation
respectively :
dU
α
ds
+
(
α
µν
)
U
µ
U
ν
= 0, (2)
D
2
Ψ
α
Ds
2
= R
α
.βγδ
U
β
U
γ
Ψ
δ
, (3)
where

α
µν

is the Christoffel symbol of the second kind R
α
βγδ
is the Riemann-
Christoffel curvature tensor.
It is worth mentioning that the Bazanski approach has been successfully applied
in geometries different from the Riemannian one [2],[3]. Also, the Lagrangian (1)
can be amended to describe path and path deviation equations of spinning charged
particles [4] by introd ucing the following Lagrangian:
L = g
αβ
U
α
DΨ
β
Ds
+ (
e
m
F
αβ
U
β
+
1
2m
R
αβγδ
S
γδ
U
α
)Ψ
µ
(4)
to give
dU
α
ds
+
(
α
µν
)
U
µ
U
ν
=
e
m
F
µ
.ν
U
ν
+
1
2m
R
α
.µνρ
S
νρ
U
µ
. (5)
Its spinning charged deviation equation becomes:
D
2
Ψ
α
Ds
2
= R
α
.µνρ
U
µ
U
ν
Ψ
ρ
+
e
m
(F
α
.ν
DΨ
ν
Ds
+ F
α
.ν;ρ
U
ν
Ψ
ρ
)
+
1
2m
(R
α
.µνρ
S
νρ
DΨ
ν
Ds
+ R
α
µνλ
S
νλ
.;ρ
U
µ
Ψ
ρ
+ R
α
µνλ;ρ
S
νλ
U
µ
Ψ
ρ
), (6)
where F
µ
.ν
is the electromagnetic field tensor and S
γδ
is the spin tensor .
1
Mathematics Department, Modern Sciences and Arts University, Giza, EGYPT
e.mail: kahil@aucegypt.edu
1

The Bazanski Approach in Brane World Models
It is well known that in the Brane world scenario our universe can be described
in terms of a 4+N dimensional, with N ≥ 1 and the 4D space-time p art of it is
embedded in a 4+N manifold [5]. Accordingly, the bulk geodesic motion is observed
by a four d imen sional observer to reproduce the physics of 4D space-time [6]. Con-
sequently, it is vital to derive the path and the path deviation equations for a test
particle on a brane u s ing the following L agrangian [7] :
L = g
µν
(x
ρ
, y)U
µ
DΨ
ν
Ds
+ f
µ
Ψ
µ
, (7)
where g
µν
(x
ρ
, y) is the induced metric andf
µ
=
1
2
U
ρ
U
σ
∂g
ρσ
∂y
dy
ds
U
µ
describes a parallel
force due to the effect of non-compactified extra dimension. The variation of the
Lagrangian gives [8]:
dU
µ
ds
+
(
µ
αβ
)
U
α
U
β
= (
1
2
U
ρ
U
σ
− g
ρσ
)
∂g
ρσ
∂y
dy
ds
U
µ
. (8)
As in the brane world models, one can express
1
2
∂g
ρσ
∂y
in terms of the extrinsic
curvature Ω
ρσ
i.e. Ω
αβ
=
1
2
∂g
ρσ
∂y
[9]. Thus, equation (8) becomes:
dU
µ
ds
+
(
µ
αβ
)
U
α
U
β
= 2(
1
2
U
µ
U
σ
− g
µσ
)Ω
ρσ
dy
ds
U
ρ
. (9)
And its corresponding deviation equ ation is
D
2
Ψ
α
Ds
2
= R
α
.µνρ
U
µ
U
ν
Ψ
ρ
+(U
α
U
σ
U
ν
)Ω
σν
∂y
dy
ds
)
;ρ
Ψ
ρ
+2((
1
2
U
α
U
σ
−g
ασ
)Ω
ρσ
dy
ds
U
ρ
)
;δ
Ψ
δ
(10)
Also, applying th e bazanski approach in brane world models, we obtain the path
and path deviation equations for a spinning charged object, respectively
dU
α
ds
+
(
α
µν
)
U
µ
U
ν
=
e
m
F
α
ν
U
ν
+
1
2m
R
α
βµν
S
µν
U
β
+ 2(
1
2
U
α
U
ρ
− g
α
ρ)Ω
ρδ
dy
ds
U
δ
(11)
and
D
2
Ψ
α
Ds
2
= R
α
.µνρ
U
µ
U
ν
Ψ
ρ
+
1
2m
(R
α
.µνρ
S
νρ
DΨ
µ
Ds
+ R
α
µνλ
S
νλ
.;ρ
U
µ
Ψ
ρ
+ R
α
µνλ;ρ
S
νλ
U
µ
Ψ
ρ
)
+
e
m
(F
α
ρ
Ψ
ρ
+ F
α
ρ
DΨ
ρ
Ds
) + (
1
2
U
α
U
σ
U
ν
)Ω
σν
dy
ds
)
;ρ
Ψ
ρ
+ (
1
2
U
α
U
σ
− g
ασ
)Ω
σν
dy
ds
DΨ
ν
Ds
.
(12)
Thus, equations (11) and (12) are derived from the following Lagrangian
L = g
µν
(x
ρ
, y)U
µ
DΨ
ν
Ds
+ 2(
1
2m
R
µνρσ
S
ρσ
U
ν
+ Ω
ρσ
∂yU
ρ
U
σ
U
µ
dy
ds
)Ψ
µ
, (13)
2

Path & Path Deviation Equations of Non-Symmetric
Geometries in Brane World Models
Path equation and Path deviation equations in Brane World Models defined in
non-symmetric geometries can be obtained by suggesting the following Lagrangian:
L = g
µν
U
µ
DΨ
ν
Dτ
+ λf
[µν]
U
µ
Ψ
ν
+
1
2
U
α
U
β
U
ρ
∂g
αβ
∂s
dy
ds
, (14)
where g
µν
= g
(µν)
+ g
[µν]
,λ is a parameter and, f
[µν]
=
ˆ
A
µ,ν
−
ˆ
A
ν,µ
is a skew
symmetric tensor related to the Yukawa force [10].
Applying the Bazanski ap proach we obtain the path equation
dU
α
ds
+
(
α
µν
)
U
µ
U
ν
= λg
αµ
f
[µν]
U
ν
+ g
ασ
g
[νσ];ρ
U
ν
U
ρ
+ (
1
2
U
ρ
U
σ
− g
ρσ
)
∂g
ρσ
∂y
dy
ds
U
µ
.
(15)
and its path deviation equation:
D
2
Ψ
α
Ds
2
= R
α
.µνρ
U
µ
U
ν
Ψ
ρ
+ 2g
σα
(g
[ν[σ];ρ]
)
DΨ
ν
Ds
U
ρ
+ λ(f
α
.ν
DΨ
ν
Ds
+ f
α
.ν;ρ
U
ν
Ψ
ρ
).
+ ((
1
2
U
α
U
ρ
− g
αρ
)
∂g
δρ
dy
U
δ
dy
ds
)
;ν
Ψ
ν
+ (
1
2
U
α
U
µ
∂g
µν
dy
dy
ds
)
DΨ
ν
Ds
(16)
The Bazanski Approach in Curved Clifford Space
It is well known that extended objects can be expressed by p-branes[11]. This type
of representation is defined in curved Clifford Space. The advantage of this space is
to show that extended objects are purely poly-geodesics satisfying a metamorphic
transformation.[12].
Thus, we suggest the following Lagrangian:
L = g
AB
P
α
∇φ
β
∇τ
+
1
2
S
αβ
∇φ
αβ
∇τ
, (17)
such that
∇φ
α
∇τ
=
dφ
α
dτ
+ Γ
α
βρ
U
ρ
φ
β
+
1
2m
ˆ
R
α
.βγδ
S
γδ
φ
β
+
1
2m
S
ω
γ
(Ξ
ασ
µω
U
µ
+
1
2m
S
ρβ
Ω
ασ
ρβω
),
where Γ
α
βρ
is the Cartan connection,
ˆ
R
α
.βγδ
is the Cartan curvature and the two other
metamorphic connections Ξ
αδ
βρ
&Ωβργ
αδ
existed due to the presence of bivector
quantities. By taking variation with respect to its deviation vector φ
δ
we get
∇U
α
∇τ
= 0 (18)
and taking the variation with respect to its deviation bivector φ
δρ
we obtain
∇S
αβ
∇τ
= 0 (19)
3

In the case of a charged particle described by a polyvector we can find that the
corresponding linear momentum equation becomes [12]
∇U
α
∇τ
=
e
m
F
α
β
U
β
(20)
and its angular momentum equation takes the followin g form
∇S
αβ
∇τ
= F
α
ν
S
νβ
− F
β
ν
S
να
(21)
We suggest the following Lagrangian to derive (20) and (21)
L = g
µν
P
µ
∇φ
ν
∇τ
+
1
2
S
µν
τφ
µν
∇τ
+ F
µν
U
ν
φ
µ
+
1
2
(F
µρ
S
ρ
ν
− F
ρ
ν
S
ρµ
)φ
µν
. (22)
If we put the flavor of Kaluza-Klein in curved C lifford space for combin ing gravity
and electromagnetism, i.e, increasing the spatial dimension by a compacted one,
then equations (20),(21) can be derived from the following Lagrangian:
L = g
AB
P
A
∇φ
B
∇ˆτ
+
1
2
S
AB
∇φ
AB
∇ˆτ
, (23)
where A = 1, 2, 3, 4, 5 to become
∇U
α
∇ˆτ
= 0 (24)
and
∇S
αβ
∇ˆτ
= 0 (25)
which means th at the path of a charged spinning particle defined in Riemannain
geometry behave like as a test particle in 5D curved Clifford space.
Discussion and Concluding Remarks
In this s tu dy, we have derived path and path deviation equations for test par-
ticles and spinn ing charged test objects in Brane World Models from one single
Lagrangian. Also, the procedure has been used to derive path and path deviation
equations for a test particle existing in non-symmetric th eory of gravity and exam-
ined how the extrinsic curvature term would be amended due to the existence of
non-symmetric terms of gravitational field. Thus, we can find that the added term
appeared in the extrinsic curvature of non-symmetric part may be connected to the
spin. Finally, we have dealt with extended objects as p-br anes defined in curved
Clifford space [13]. This type of space has a deeper understanding of physics, i.e.,
quantities in nature could be defined by polyvectors. From this pers pective we have
developed its correspon ding Bazanski Lagrangian to include deviation bi-vectors to-
gether with deviation vectors. The importance of this approach is that we can derive
from one Lagrangian(17) two simultaneous quantities responsible for conservation
of momentum (18) and angular momentum (19) respectively. Using this mecha-
nism, we have found that the Dixon-Souriau equations in Riemannian geometry
could be seen like equations for ch arged object and their angular momentum part
4

defined in curved clifford spaces. In addition, if we have the flavor of Kaluza-Klein
of unifying gravity and electromagnetism in curved Clifford space, we must increase
the dimension of this space by an extra compacted dimension to pr eserve the con-
servation of charges. Consequ ently, from equations (24) and (25) we can find that
charged spinning particles behave like test particles defined in higher dimensional
curved clifford space. To conclude this study, we must take into consideration that
unification processes can be achieved if we increase the number of dimensions and
extend the geometries to geometrize all quantities that appeared in our approach .
Acknowlgements
The author would like to thank Professors M.I. Wanas, M. Abdel-Megied , G.S.
Hall, T. Harko, G. De Young and Mr. W.S. El-Hanafy for their useful comments.
A word of thanks should be addressed to Professors A. Rajantie and F.H. Stoica
for their su ppport and help to pariticapate in PAS COS07. A very special word of
thankfulness should be sent to Dr. N. El-Degwi and Professor K . Abdel Hamid for
helping me to get a grant to represent MSA University at this confenence.
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